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Source. Published pp. 3–4, Lemma 3.1 and its proof.

Statement. For every integer $n\ge11$, some $\epsilon_n>0$ has this property: any real array $b_{ij}$, $1\le i<j\le n$, satisfying $|b_{ij}-1|<\epsilon_n$ is realized as the squared pair distances of $n$ vertices in a brick of dimension $\binom n2$, with every edge length positive. No prior realizability assumption on the array is necessary.

Proof. Index coordinates by pairs $\{j,k\}\subseteq[n]$. For positive numbers $x_{jk}$, let the proposed vertex $x^{(i)}$ have coordinate $x_{jk}$ when $i\in\{j,k\}$ and zero otherwise. It is a vertex of the brick with coordinate edge lengths $x_{jk}$. Put $z_{jk}=x_{jk}^2=z_{kj}$. The required squared distances are precisely

$$\sum_{l\ne i,j}(z_{il}+z_{jl})=b_{ij}.$$

Let $M$ be this square system's $\binom n2$ by $\binom n2$ coefficient matrix. The row indexed by $\{i,j\}$ is the incidence vector of

$$F_{ij}=\{\{k,l\}:|\{i,j\}\cap\{k,l\}|=1\}.$$

Each row has $2(n-2)$ ones. For pairs sharing one vertex, say $ij$ and $ik$, the common members are the $n-3$ edges from $i$ to another vertex, together with $jk$, so $|F_{ij}\cap F_{ik}|=n-2$. For disjoint pairs $ij$ and $kl$, the common members are $ik,il,jk,jl$, so the intersection size is four.

Take $a=2n-4$, $b=n-2$, $m=n-6$. The two intersection sizes are congruent modulo $m$, but $a-b=n-2\equiv4\not\equiv0\pmod{n-6}$ for $n\ge11$. The fully proved modular independence lemma therefore makes the rows independent over $\mathbb Q$. Since $M$ is a square integer matrix, its determinant is nonzero and it is invertible over $\mathbb R$.

For the all-ones array, the solution is $z_{ij}=1/[2(n-2)]>0$. The solution $z=M^{-1}b$ varies continuously with the finite vector $b$. Choose a sufficiently small neighborhood of the all-ones array so every coordinate of $z$ stays positive; then set $x_{ij}=\sqrt{z_{ij}}$. The displayed equations give every required distance and complete the proof.

Scope. The source selects this combinatorial invertibility argument; it is retained here with its quoted input expanded. No optimal value of $\epsilon_n$ or classification of the smaller values of $n$ is asserted. The source's warning about $n=4$ concerns this lemma, not the failure of all four-point configurations to be Ramsey.

Bears on. #174.